It’s A Math, Math World (Power/Smpl Size I)

In this post, we will examine type I and type II errors and their relation to sample size and power calculations.

We start with a few definitions:

In a clinical trial, there are 2 types of error that we want to control for:

  • Type I error (False Positive or Consumer’s Risk): This is a decision that finds that the new treatment works better when in fact it really does not.

This error rare is controlled by FDA or other regulatory agencies. Depending on setting, α = 0.05, 0.01 or 0.001 might be required.

  • Type II error (False Negative or Producer’s Risk): This is a decision that fails to find that the treatment works better when in fact it does.

This error rate is controlled more by the company. They have more say in setting this rate, but an irresponsible type II error rate will adversely influence drug approval. For research, a type II error β = 0.20 is usually adequate.

Power = 1 – Type II Error: The chance to detect a difference when one exists.

If there is no bias, then the quality of the study is directly proportional to the sample size.

  • If you have more subjects, then the smaller the error of the estimates and the better the type I and type II errors.
  • IF sample size is too small, then, given type I error is maintained, effective therapy may not be discovered.
  • If sample size is too large, then the study is too expensive and difficult to be done.


It is important to either:

  • Find the minimum sample size to obtain a specified power.
  • Determine the specific power for a given sample size.

However there are many formulas for power and sample size for different:

Outcome types:

  • Continuous
  • Proportions
  • Survival data

Trial purpose:

  • Superiority vs. Non-equivalency

Design of Trial:

  • Matched vs. unmatched study
  • Cluster vs. independent sampling
  • Adjusted for covariates vs. unadjusted analysis

Next time, we will look at specific examples of power calculations.

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13 Responses to “It’s A Math, Math World (Power/Smpl Size I)”

  • The only jarring thing to me was that you have the producers’ risk as type II and consumers’ risk as type I. Perhaps I am aging, but all my OLD books give it the other way around. Other than that, I am delighted to see the subject of power being reintroduced into the world of polite conversation.

  • monica chaudhari says:

    I liked this blog very much. It is a great way of disseminating knowledge.I wish that at some point we also talk about estimation of power and sample size in cases of sequential analysis wherein decision based on first stage is likely to effect subsequent stages.

  • Thomas Erdahl says:

    I heard about this through Linked-In, and I hope you continue this!
    Thanks and regards,

  • Michelle says:

    Thank you for the excellent refresher of my graduate level statistics.

  • Rose Somers, M.D. says:

    I have been asking for so long, exactly how do you figure out sample size for a trial; that is, how many subjects to enroll? I understand what it is based on and why, but not how to get to the number! I guess I’ll have to wait for next episode to find out. Could you give detailed examples of different types of trials and your caculations of sample size needed to detect the expected effect? Thank you very much.

  • admin says:

    Dr. Somers, that is exactly my plan in the next week or so. I will try to make it a short wait! Thank you for taking the time to read my blog.

  • admin says:

    You are welcome. Glad to help out.

  • admin says:

    Thank you. Have you read the other posts? I should be posting new in about a week or so.

  • admin says:

    I wil see what I can do.

  • admin says:

    Thanks, Robert, glad to be of service! BTW, I will double check my notes about the “risk” issue to make sure that I did not get it backwards! THank you for your observations and please enjoy my past postings as you await my new post that should come out very soon.

  • admin says:

    I am planning to do so soon. stay tuned…..

  • Scott Malinowski, Pharm.D. says:

    Great review! I discuss this concept with my pharmacy students on a frequent basis, and am always looking for good, plain, non-statistician, terminology to get the point across.

    In regards to Robert’s comment, I think you got it right: the field (scientific discipline, FDA, etc) will determine which alpha is considered appropriate to use – this will also be vetted by the grant/manuscript peer review process. As far as beta error, to some degree the field influences this also (by determining which level of error/power is acceptable for publication of a study, and the peer review process), but the researcher ultimately determines which sample size to use in the study – which should be calculated from a power analysis (which factors in effect size, alpha and beta error, etc). As you mentioned, logistics and funding also play a part. When critiquing studies, students always comment “they should have used a larger sample size” – easier said than done, I proceed to explain.

    Looking forward to more posts!

  • C.S. Ganti says:

    We are connected and corresponded via Linked IN. This is a very important site and has an excellent value. I am quite familiar (while my education is some time back) with some of the techniques you are presenting in your blog.

    DOE – as abbreviated is now also find its way in Marketing type applications. I know your specialty is in Pharmaceutical / Clinical areas.

    Thank you,
    C.S. Ganti